I have seen it done, and it will work after a fashion. The .30-'06 reamer won't clean up the chamber because the .308 is wider at the shoulder than the .30-'06 is at that point, but all that happens is that the fired case has a bulge around it behind the shoulder. Not good for many reloads, maybe, but the owner of the rifle in question didn't reload and didn't care. (For that rifle, the .308 action was the same as the .30-'06 action, so that was not a factor.)
So I can't say it won't work, only that I too don't recommend it and don't see much point in it.
Jim